CHAPTER
4
Probability MULTIPLE CHOICE Section 4.1 1.
A statistical experiment is a process that, when performed: a. results in one and only one of two observations b. results in at least two of many observations c. may not lead to the occurrence of any outcome d. results in one and only one of many observations
2. An outcome is: a. what is observed when an experiment is performed b. what happens when you do not perform an experiment c. a collection of many events d. a collection of at least two sample spaces 3. A sample point is: a. a collection of many sample spaces b. a point that represents a population in a sample c. an element of a sample space d. a collection of observations 4. The performance of an experiment results in: a. one and only one observation b. many observations c. exactly two observations d. one of the two observations 5. An event: a. is the same as a sample space b. includes exactly one outcome c. includes one or more of the outcomes d. includes all possible outcomes 6. A simple event: a. is a collection of exactly two outcomes b. includes one and only one outcome c. does not include any outcome d. includes all possible outcomes 7. A compound event includes: a. at least three outcomes b. at least two outcomes c. one and only one outcome d. all outcomes of an experiment 79
8. The experiment of tossing a coin twice has: a. two outcomes b. three outcomes c. four outcomes d. eight outcomes 9. The experiment of rolling a die once and observing for an even or odd number has: a. six outcomes b. eight outcomes c. four outcomes d. two outcomes 10. Two households are randomly selected and it is observed whether or not each of them owns a telephone answering machine. The total number of outcomes for this experiment is: a. six b. eight c. two d. four 11. A box contains a few red and a few white marbles. Two marbles are randomly drawn from this box and the color of these marbles is observed. The total number of outcomes for this experiment is: a. eight b. four c. six d. two 12.
You randomly select two households and observe whether or not they own a telephone answering machine. Which of the following is a simple event? a. Exactly one of them owns a telephone answering machine. b. At least one of them owns a telephone answering machine. c. At most one of them owns a telephone answering machine. d. Neither of the two owns a telephone answering machine.
13. A box contains a few red and a few white marbles. After randomly drawing two marbles from this box, you observe their color. Which of the following is an example of a simple event? a. At most one marble is red. b. At least one marble is white. c. Both marbles are white. d. Not more than one marble is red. 14. You toss a coin twice and observe two tails. This event is a: a. compound event b. simple event c. multiple outcome d. multinomial sample point
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Chapter 4 Probability 15. A researcher samples two adults to determine if they favor or oppose gun control. Assume that there are only two outcomes for each adult—either he/she favors or he/she opposes gun control. Which of the following is a compound event? a. Both adults favor gun control. b. Exactly two adults favor gun control. c. Neither of the two opposes gun control. d. At most one favors gun control. 16. A box contains a certain number of auto parts. A few of these parts are defective and the remaining are good. You select two parts at random from this box and observe if they are good or defective. Which of the following is a compound event? a. At most one part is defective. b. Both parts are good. c. Neither one of the two parts is good. d. No compound event can be defined.
Section 4.2 17. The numerical measure of the likelihood that a specific event will occur is: a. the sample space b. a sample point c. an event d. the probability of an event 18. The probability of an event is always: a. greater than zero b. less than 1 c. in the range zero to 1 d. greater than 1 19. The sum of the probabilities of all final outcomes for an experiment is always: a. equal to 1 b. equal to zero c. less than 1 d. greater than 1 20. We apply the classical probability approach to an experiment that: a. cannot be repeated b. has equally likely outcomes c. does not have more than two outcomes d. has all independent outcomes 21. We apply the relative frequency probability approach to experiments that: a. do not have equally likely outcomes but can be repeated b. do not have equally likely outcomes and cannot be repeated c. have equally likely outcomes and cannot be repeated d. have equally likely outcomes and can be repeated
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22. We apply the subjective probability approach to experiments that: a. do not have equally likely outcomes and cannot be repeated b. do not have equally likely outcomes but can be repeated c. have equally likely outcomes but cannot be repeated d. have equally likely outcomes and can be repeated 23. According to the classical probability rule, the probability of a simple event is: a. the total number of outcomes for the experiment divided by 1 b. 1 divided by the sample space c. 1 divided by the total number of outcomes for the experiment d. 1 divided by the compound event 24. According to the classical probability rule, the probability of a compound event is: a. the total number of outcomes for the experiment divided by 4 b. the number of outcomes favorable to the given event divided by the total number of outcomes c. 5 divided by the total number of outcomes for the experiment d. 2 divided by the number of outcomes favorable to the given event 25. According to the relative frequency concept of probability, the probability of an event is: a. 1 divided by the total number of outcomes for the experiment b. the number of times the given event is observed divided by the total number of repetitions of the experiment c. the number of outcomes favorable to the given event divided by the sample space d. the sample space divided by the number of outcomes favorable to the given event 26. If you roll a die once, the probability of obtaining an odd number is: a. .60 b. .25 c. .17 d. .50 27. You select one person from a group of eight males and two females. The two events ―a male is selected‖ and ―a female is selected‖ are: a. independent b. equally likely c. not equally likely d. collectively exhaustive 28. Which of the following values cannot be the probability of an event? a. .78 b. .00 c. 1.25 d. 1.00 29. Which of the following values cannot be the probability of an event? a. .91 b. .00 c. 1.00 d. –.56 82
Chapter 4 Probability 30. A research firm polls 25 persons to determine their opinions on income tax reforms. The probabilities that a person is in favor of income tax reforms, he/she is against it, and he/she has no opinion should all add up to: a. .00 b. .85 c. 1.00 d. more than 1.0 31. In a group of 80 students, 16 are seniors. If you select one student randomly from this group, the probability that this student is a senior is: a. .80 b. .16 c. .20 d. .25 32. In a group of 400 families, 300 own houses. If you select one family randomly from this group, the probability that this family owns a house is: a. .25 b. .75 c. .80 d. .40 33. You roll an unbalanced die 500 times, and a 3-spot is obtained 110 times. The probability of obtaining the 3-spot for this die is approximately: a. .92 b. 1/6 c. .22 d. .13 34. A quality control staff selects 200 items from the production line of a company and finds 14 of them defective. The probability that an item manufactured by this company is defective is approximately: a. .07 b. .14 c. .04 d. .19
Sections 4.3-4.8 35. The marginal probability is the probability of: a. a sample space b. an outcome when another outcome has already occurred c. an event without considering any other event d. an experiment calculated at the margin 36. The conditional probability is the probability: a. of a sample space based on a certain condition b. that an event will occur given that another event has already occurred c. that an event will occur based on the condition that no other event is being considered d. that an event will occur based on the condition that no other event has already occurred
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37. The conditional probability of event A given that event B has already occurred is: a. P(A or B) b. P(Bďƒ´A) c. P(A and B) d. P(Aďƒ´B) 38. Two mutually exclusive events: a. always occur together b. can sometimes occur together c. cannot occur together d. can occur together, provided one has already occurred 39. Two events are independent if the occurrence of one event: a. affects the probability of the occurrence of the other event b. does not affect the probability of the occurrence of the other event c. means that second event cannot occur d. means that second event is definite to occur 40. Two events are dependent if the occurrence of one event: a. affects the probability of the occurrence of the other event b. does not affect the probability of the occurrence of the other event c. means that second event cannot occur d. means that second event is definite to occur 41. Two independent events are: a. always mutually exclusive events b. never mutually exclusive events c. always complementary events d. always subjective events 42. Two mutually exclusive events are always: a. independent events b. complementary events c. dependent events d. subjective events 43. Two complementary events: a. have no common outcomes b. can have common outcomes c. contain the same outcomes d. are always subjective events 44. Two complementary events: a. taken together do not include all outcomes for an experiment b. taken together include all outcomes for an experiment c. can occur together d. are always independent
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Chapter 4 Probability 45. Two events A and B are independent if: a. P(A) is equal to P(B) b. P(BA) is equal to P(A) c. P(AB) is equal to P(A) d. P(AB) is equal to P(B) 46. If P(AB) is equal to P(A), then events A and B are: a. complementary events b. mutually exclusive events c. independent events d. subjective events Use the following information for questions 47 –53: The following table gives the two-way classification of 500 students based on sex and whether or not they suffer from math anxiety. Sex Male Female
Suffer From Math Anxiety Yes No 160 80 175 85
47. If you randomly select one student from these 500 students, the probability that this selected student is a female is: a. .480 b. .653 c. .520 d. .347 48. If you randomly select one student from these 500 students, the probability that this selected student suffers from math anxiety is: a. .670 b. .275 c. .333 d. .330 49. If you randomly select one student from these 500 students, the probability that this selected student suffers from math anxiety, given that he is a male is approximately: a. .333 b. .673 c. .327 d. .667 50. If you randomly select one student from these 500 students, the probability that this selected student is a female, given that she does not suffer from math anxiety is approximately: a. .649 b. .327 c. .889 d. .515
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51. Events ―Yes‖ and ―No‖ are: a. mutually exclusive events b. mutually nonexclusive events c. subjective events d. conditional events 52. Which of the following events are mutually exclusive? a. Male and yes b. Female and no c. Yes and no d. No and male 53. Which of the following events are mutually nonexclusive? a. No and yes b. Female and no c. Female and male d. Female and yes Use the following information for questions 54 –60: A pollster asked 1,000 adults whether Republicans or Democrats have better domestic economic policies. The following table gives the two-way classification of their opinions. Sex Male Female
Republicans 220 170
Democrats 340 200
No Opinion 40 30
The pollster then randomly selected one adult from these 1,000 adults. 54. The probability that the selected adult is a male is: a. .400 b. .550 c. .600 d. .450 55. The probability that the selected adult says Democrats have better domestic economic policies is: a. .390 b. .540 c. .070 d. .400 56. The probability that the selected adult is a female given that she thinks that Republicans have better domestic economic policies is approximately: a. .579 b. .382 c. .686 d. .436
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Chapter 4 Probability 57. The probability that the selected adult has no opinion given that he is a male is approximately: a. .667 b. .375 c. .067 d. .075 58. Events ―Republican‖ and ―No opinion‖ are: a. conditional events b. mutually nonexclusive events c. subjective events d. mutually exclusive events 59. Which of the following events are mutually exclusive? a. Male and no opinion b. Female and Democrats c. Democrats and no opinion d. Republican and male 60. Which of the following events are mutually nonexclusive? a. No opinion and male b. Female and male c. Democrats and Republicans d. No opinion and Republicans 61. The probability that a randomly selected college student is a part-time student is .18. The probability of the complementary event of this event is: a. .18 b. .64 c. .82 d. cannot find 62. The probability that a family owns stocks is .56. The probability of the complementary event of this event is: a. .65 b. .44 c. .25 d. cannot find 63. The intersection of two events A and B represents the outcomes that are: a. either in A or in B or in both A and B b. common to both A and B c. either in A or in B d. not common to both A and B 64. The joint probability of two events A and B is that: a. either event A happens or event B happens b. neither of the events A and B happens c. both events A and B happen d. both events A and B happen less either event A or B has already happened.
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65. The intersection of two events A and B is written as: a. (A or B) b. (A and B) c. (AB) d. (BA) 66. The probability of the intersection of two events A and B is given by: a. P(A) + P(B) b. P(A) + P(B) – P(A and B) c. P(A) P(AB) d. P(A) P(BA) 67. The joint probability of two independent events A and B is: a. P(A) + P(B) b. P(A) + P(B) + P(A or B) c. P(A) P(B) d. P(A) P(AB) 68. The joint probability of two mutually exclusive events is always equal to: a. 1 divided by the number of outcomes b. .50 c. zero d. 1.0 Use the following information for questions 69 –73: The following table gives a two-way classification of all employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies.
Smoker Nonsmoker
Suffer From Allergies Yes No 35 25 55 185
69. The joint probability of events ―smoker‖ and ―no‖ is approximately: a. .833 b. .459 c. .162 d. .083 70. The joint probability of events ―yes‖ and ―nonsmoker‖ is approximately: a. .444 b. .617 c. .067 d. .183 71. The joint probability of events ―yes‖ and ―no‖ is: a. .50 b. 1.00 c. .00 d. .24 88
Chapter 4 Probability 72. The probability that a randomly selected employee is a smoker and suffers from allergies is approximately: a. .011 b. .351 c. .117 d. .083 73. The probability that a randomly selected employee is a nonsmoker and does not suffer from allergies is approximately: a. .617 b. .008 c. .500 d. .281 Use the following information for questions 74 –79: A pollster asked 1,000 adults whether Republicans or Democrats have better domestic economic policies. The following table gives the two-way classification of their opinions. Sex Male Female
Republicans 220 170
Democrats 340 200
No Opinion 40 30
74. The joint probability of events ―Republicans‖ and ―Male‖ is: a. .170 b. .340 c. .220 d. .880 75. The joint probability of events ―Female‖ and ―Democrats‖ is: a. .200 b. .500 c. .340 d. .660 76. The joint probability of events ―Male‖ and ―No Opinion‖ is: a. .040 b. .030 c. .300 d. .400 77. The joint probability of events ―Democrats‖ and ―No Opinion‖ is: a. .340 b. .000 c. .040 d. .030
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78. The probability that a randomly selected adult from these 1000 adults is a female and holds the opinion that Republicans have better domestic economic policies is: a. .022 b. .017 c. .220 d. .170 79. The probability that a randomly selected adult from these 1000 adults is a male and holds the opinion that Democrats have better domestic economic policies is: a. .200 b. .220 c. .600 d. .340 80. In a class of 40 students, 10 are math majors. The teacher selects two students at random from this class. The probability that both of them are math majors is approximately: a. .063 b. .937 c. .250 d. .058 81. The athletic department of a school has 12 full-time coaches, and four of them are female. The director selects two coaches at random from this group. The probability that neither of them is a female is approximately: a. .091 b. .445 c. .639 d. .424 82. The probability that a physician is a pediatrician is .20. The administration selects two physicians at random. The probability that none of them is a pediatrician is: a. .04 b. .16 c. .64 d. .80 83. The probability that an adult possesses a credit card is .70. A researcher selects two adults at random. The probability that the first adult possesses a credit card and the second adult does not possess a credit card is: a. .57 b. .92 c. .21 d. .35 84. The probability that a person is a college graduate is .35 and that he/she has high blood pressure is .12. Assuming that these two events are independent, the probability that a person selected at random is a college graduate and has high blood pressure is: a. .455 b. .801 c. .095 d. .042 90
Chapter 4 Probability 85. The probability that a corporation made profits in 2003 is .70 and the probability that a corporation made charitable contributions in 2003 is .25. Assuming that these two events are independent, the probability that a corporation made profits in 2003 and made charitable contributions in 2003 is: a. .075 b. .175 c. .525 d. .225 86. The probability that an employee of a company is a male is .60 and the joint probability that an employee of this company is a male and single is .18. The probability that a randomly selected employee of this company is single given he is a male is: a. .400 b. .820 c. .300 d. .108 87. The probability that a farmer is in debt is .70. The joint probability that a farmer is in debt and also lives in the Midwest is .28. The probability that a randomly selected farmer lives in the Midwest, given that he is in debt is: a. .196 b. .300 c. .720 d. .400 88. The total number of outcomes for three rolls of a die is: a. 216 b. 36 c. 8 d. 1296 89. A woman owns 12 blouses, seven skirts, and six pairs of shoes. She will randomly select one blouse, one skirt, and one pair of shoes to wear on a certain day. The total number of possible outcomes is: a. 84 b. 504 c. 72 d. 42
Section 4.9 90. The union of two events A and B represents the outcomes that are: a. either in A or in B or in both A and B b. common to both A and B c. neither in A nor in B d. not common to both A and B 91. The probability of the union of two events A and B is the probability that: a. neither event A happens nor event B happens b. both events do not happen together c. both events A and B happen together d. either event A or event B or both A and B happen 91
92. The union of two events A and B is written as: a. (A or B) b. (A and B) c. (AB) d. (AB) 93. The probability of the union of two events A and B is: a. P(A) + P(B) + P(A and B) b. P(A) + P(B) – P(A and B) c. P(A) P(AB) d. P(A) P(BA) 94. The probability of the union of two events A and B, that are mutually exclusive, is: a. P(A) + P(B) b. P(A) – P(B) + P(A and B) c. P(A) P(AB) d. P(A) P(BA) Use the following information for questions 95 –98: A pollster asked 1,000 adults whether Republicans or Democrats have better domestic economic policies. The following table gives the two-way classification of their opinions. Sex Male Female
Republicans 220 170
Democrats 340 200
No Opinion 40 30
95. If the pollster randomly selects an adult from these 1,000 adults, the probability that this adult has no opinion is: a. .046 b. .390 c. .070 d. .400 96. If the pollster randomly selects one adult from these 1,000 adults, the probability that this adult is a female or thinks that Democrats have better domestic economic policies is: a. .940 b. .200 c. .740 d. .400 97. If the pollster randomly selects one adult from these 1,000 adults, the probability that this adult has no opinion or is a male is: a. .670 b. .070 c. .600 d. .630
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Chapter 4 Probability 98. If the pollster randomly selects one adult from these 1,000 adults, the probability that this adult is a male or thinks that Republicans have better domestic economic policies is: a. .770 b. .990 c. .390 d. .600 Use the following information for questions 99 –101: A consumer researcher inspects 300 batteries manufactured by two companies for being good or defective. The following table gives the two-way classification of these 300 batteries. Company A Company B
Good 140 130
Defective 10 20
99. If the researcher selects one battery at random from these 300 batteries, the probability that this battery is either good or made by company B is approximately: a. .900 b. .433 c. .967 d. .420 100. If the researcher selects one battery at random from these 300 batteries, the probability that this battery is either Company A’s or defective is approximately: a. .567 b. .033 c. .600 d. .820 101. If the researcher selects one battery at random from these 300 batteries, the probability that this battery is either company A’s or good is approximately: a. .420 b. .067 c. .900 d. .933 102. The probability that a student at a university is a male is .48, that a student is a business major is .14, and that a student is a male and a business major is .06. The probability that a randomly selected student from this university is a male or a business major is: a. .62 b. .56 c. .68 d. .40 103. The probability that a family has at least one child is .74, that a family owns a camcorder is .25, and that a family has at least one child and owns a camcorder is .13. The probability that a randomly selected family has at least one child or owns a camcorder is: a. .99 b. .86 c. .87 d. .62 93
104. Forty-five percent of the voters are in favor of limiting the number of terms for senators and congressmen, 36% are against it, and 19% have no opinion. If a pollster selects one voter at random, the probability that this voter is either in favor of limiting the number of terms for senators and congressmen or has no opinion is: a. .62 b. .28 c. .64 d. .55 105. A company has a total of 500 male employees. Of them, 125 are single, 280 are married, 65 are either divorced or separated, and 30 are widowers. If management selects one male employee at random from the company, the probability that this employee is married or a widower is: a. .310 b. .375 c. .250 d. .620 106. The probability that a person is a college graduate is .55 and that he/she has high blood pressure is .20. Assuming that these two events are independent, the probability that a person selected at random is a college graduate or has high blood pressure is: a. .11 b. .64 c. .75 d. cannot be calculated from the given information 107. The probability that a corporation made profits in 2003 is .70 and the probability that a corporation made charitable contributions in 2003 is .25. Assuming that these two events are independent, the probability that a corporation made profits in 2003 or made charitable contributions in 2003 is: a. .775 b. .950 c. .175 d. cannot be calculated from the given information
Chapter Review Questions 108. Three football fans are asked whether they think that the Tampa Bay Buccaneers will repeat as NFL champions in the 2003-04 season. The only possible responses are ―Yes‖ or ―No‖. The first person asked lives in Tampa Bay and always chooses the Buccaneers to win the championship. Knowing this, how many possible outcomes are there? a. 4 b. 8 c. 2 d. 3 109. If you toss a coin three times, one possible outcome is ―head, head, head‖. The event ―two heads, one tail‖ is a(n): a. impossible event b. sure event c. compound event d. simple event 94
Chapter 4 Probability 110. When rolling a pair of fair dice, what is the probability that the sum of the dice on one roll will be seven? a. 1/6 b. 1/12 c. 1/36 d. 5/36 111. A particular shoe factory has two machines. Machine A manufactures shoes for the left foot, and machine B manufactures shoes for the right foot. Four of 100 shoes from machine A are found to be defective, and five of 100 shoes from machine B are found to be defective. What is the probability that the next left shoe and the next right shoe will both be defective? a. .09 b. .002 c. .025 d. .2 112. A veterinarian conducts an experiment with 50 dogs. The vet gives each dog a dish of dry dog food and a dish of canned dog food at the same time. He judges that the food the dog eats first is the ―preferred‖ food. He groups the dogs as small (under 40 lbs.), medium (40–100 lbs.), and large (over 100 lbs.). The results appear as follows: Small Dogs Medium Dogs Large Dogs
Preferred Dry 2 7 4
Preferred Canned 18 13 6
What is the probability that a dog chosen at random from the group will be large and will prefer canned food? a. 6/10 b. 6/37 c. 4/50 d. 6/50 113. You roll four fair, unweighted dice. What is the probability that all 4 dice show the same number? a. 1/24 b. 1/216 c. 1/1296 d. 1/6 114. A boy is playing an adventure game. At one point, he has to make a decision to go right or go left. If he goes right, the probability that he will ―die‖ is .30. If he goes left, the probability of ―death‖ is .40. He has an equal probability of choosing either direction. What is the probability that he will ―die‖ after making his decision? a. .12 b. .70 c. .40 d. .35
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115. All 400 employees at an accounting firm indicated what size of vehicle they drive: compact car, mid-size car, large car, or truck. The following table gives a two-way classification of their responses. Male Female
Compact Car 41 52
Mid-Size Car 82 75
Large Car 24 38
Truck 49 49
Management selects one employee at random. What is the most likely outcome? a. The employee is a male and drives a mid-size car b. The employee is a female and drives a mid-size car c. The employee is a male and doesn’t drive a mid-size or large car d. The employee is a female and doesn’t drive a compact or mid-size car 116. A game requires a fair die and bag containing seven white marbles and three black marbles. The player rolls the die, and if the number on the die is above four, then the player may draw from the bag of marbles. The player wins if he/she draws a black marble from the bag. If the number on the die is four or less, the player loses without drawing from the bag. What is the probability of winning the game? a. .33 b. .30 c. .63 d. .10 117. The probability that a randomly selected person from a certain community has been on the Internet and can point out the location of Guinea on a world map is .08, and the probability that the person can point out Guinea given that he/she has been on the Internet is .20. What is the probability that a person in the community has been on the Internet? a. .40 b. .016 c. .80 d. 2.50
Multiple Choice Summary Table Item 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Answer d a c a c b b c d d b d c b d a
Item 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
Answer b d c c d c c b c a c b d c b a
Item 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64.
Answer d d a c b c b d c d c a c b b c 96
Item 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88.
Answer a c a a b d d d d c c d b c d a
Item 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112.
Answer d a c a d b b c d b a a c a b d
Chapter 4 Probability 17. 18. 19. 20. 21. 22. 23. 24.
d c a b a a c b
41. 42. 43. 44. 45. 46. 47. 48.
b c a b c c c a
65. 66. 67. 68. 69. 70. 71. 72.
b d c c d d c c
89. 90. 91. 92. 93. 94. 95. 96.
b a d a b a c c
113. 114. 115. 116. 117.
b d c d a
PROBLEM AND ESSAY Section 4.1-4.3 1. You randomly select two households and observe whether or not they own VCRs. a. How many total outcomes are possible for this experiment? Draw a tree diagram and a Venn diagram for this experiment. b. List all the outcomes included in each of the following events and indicate which are simple and which are compound events? (i) at least one household owns a VCR (ii) at most one household owns a VCR (iii) both households own a VCR (iv) the first household owns a VCR and the second household does not own a VCR 2. A box contains a few pink and a few black marbles. You randomly draw two marbles from this box and the color of these marbles is observed. a. How many total outcomes are possible for this experiment? Draw a tree diagram and a Venn diagram for this experiment. b. List all the outcomes included in each of the following events and indicate which are simple and which are compound events? (i) at most one marble is pink (ii) not more than one marble is black (iii) none of the marbles is pink (iv) the first marble is black and the second marble is pink 3. What are the two properties of probability? Give a brief explanation of each of those. 4. In a group of 200 households, 124 own telephone answering machines. Select randomly one household from this group. What is the probability that this household owns a telephone answering machine? 5. You toss an unbalanced 900 times and observe a head 540 times. What is the approximate probability of observing a head for this coin? 6. A multiple-choice question on a test has five answers. If a student randomly selects one answer from these five, what is the probability that the selected answer is correct? 7. In a total of 400 employees of a company, 280 are nonsmokers. Management selects randomly one employee from this company. What is the probability that this employee is a nonsmoker? 8. Of the 1,200 babies born at a hospital during the past five years, 660 were girls. What is the approximate probability that the next baby born at this hospital will be a girl? 97
Section 4.4 9. The following table gives the two-way classification of 400 students based on sex and whether or not they work while being full-time students. Male Female
Work 120 130
Do Not Work 60 90
a. Select one student randomly from this group of 400 students. What is the probability that this student: (i) does not work (ii) is a female (iii) does not work given he is a male (iv) is a female given she works b. Are the events ―male‖ and ―do not work‖ mutually exclusive? Explain why or why not. c. Are the events ―female‖ and ―do not work‖ independent? Explain why or why not. d. What is the complementary event of the event ―do not work‖? What is the probability of this complementary event? 10. An independent research team inspects 300 batteries manufactured by two companies for being good or defective. The following table gives the two-way classification of these 300 batteries. Company A Company B
Good 140 130
Defective 10 20
a. The team selects one battery randomly from these 300 batteries. Find the probability that this battery: (i) is manufactured by company B (ii) is defective (iii) is good given that it is manufactured by company B (iv) is manufactured by company A given that it is defective b. Are the events ―company A‖ and ―defective‖ mutually exclusive? Explain why or why not. c. Are the events ―good‖ and ―company A‖ independent? Explain why or why not. d. What is the complementary event of the event ―defective‖? What is the probability of this complementary event?
Section 4.5-4.9 11. There are a total of 300 professors at a university. Of them, 75 are female and 90 are professors in the social sciences. Of the 75 females, 30 are professors in social sciences. Are the events ―female‖ and ―professor in social sciences‖ independent? Are they mutually exclusive? Explain why or why not? 12. There are a total of 40 students in a class. Of them, 18 are male and 14 are seniors. Of the 18 males, six are seniors. Are the events ―male‖ and ―senior‖ independent? Are they mutually exclusive? Explain why or why not. 13. Let A be the event that a randomly selected family owns a house. The probability of event A is .68. What is the complementary event of A, and what is its probability?
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Chapter 4 Probability 14. Let A be the event that a randomly selected employee of a company is in favor of labor unions. The probability of event A is .56. What is the complementary event of A, and what is its probability? 15. The following table gives a two-way classification of all employees of a company based on their sex and whether or not they are college graduates. Sex Male Female
College Graduate Yes No 35 50 25 40
Management selects one employee randomly from the company. Find the probabilities: a. P(female and college graduate) b. P(male and not a college graduate) 16. A medical research unit asks 500 people whether or not they visited their physicians’ offices during the last year. The following table gives a two-way classification of their responses. Sex Male Female
Visited Physician’s Office Last Year Yes No 210 90 160 40
If the unit randomly selects one person from this group, find the following probabilities. a. P(male and visited his physician’s office) b. P(female and did not visit her physician’s office) 17.
The following table gives a two-way classification of 1000 couples based on whether one or both spouses work and whether or not they have children. Have Children Yes No 140 260 380 220
Work Status Both Spouses Work Only One Spouse Works
Select one couple randomly from these 1000 couples and find the following probabilities. a. P(both spouses work and have no children) b. P(only one spouse works and have children) 18. A university has 250 faculty members. Of them, 50 are female. You select two faculty members at random from these 250 faculty members. Find the probability that both of them are females. 19. An insurance company isolates a group of 100 car owners, 40 own foreign cars. The company selects two persons at random from these 100 persons. Find the probability that neither of them owns a foreign car. 20. Forty percent of banks did not earn profits during 2003. The SEC selects two banks at random. Find the probability that neither of them earned profits in 2003.
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21. The probability that a person has blood type A is .30. The Red Cross selects two persons at random. Find the probability that the first person has blood type A and the second person does not have blood type A. 22. The probability that a person drinks at least five cups of coffee a day is .25, and the probability that a person has a high blood pressure is .10. Assuming that these two events are independent, find the probability that a person selected at random drinks less than five cups of coffee a day and has a high blood pressure. 23. The probability that a family owns a house is .65 and that a family owns a camcorder is .12. Assuming that these two events are independent, find the probability that a family selected at random owns a house but does not own a camcorder. 24. The probability that a faculty member at a large university is a female is .30. The joint probability that a faculty member is a female and holds a doctoral degree is .24. Find the probability that a randomly selected faculty member from this university holds a doctoral degree given she is a female. 25. The probability that a family owns a VCR is .60. The joint probability that a family owns a VCR and a camcorder is .42. Find the probability that a randomly selected family owns a camcorder given that this family owns a VCR. 26. How many different outcomes are possible for five tosses of a coin? 27. The provost decides to form a committee by selecting one of 10 professors, one of 20 students, and one of six administrators. How many of outcomes are possible? 28. The following table gives the two-way classification of 400 students based on whether or not they work while being full-time students. Male Female
Work 120 130
Do Not Work 60 90
You select one student at random from these 400 students. Find the following probabilities. a. P(female or does not work) b. P(works or male) 29. The following table gives a two-way classification of all employees of a company based on their sex and whether or not they are college graduates. Sex Male Female
College Graduate Yes No 35 50 25 40
Management selects one employee randomly from the company. Find the following probabilities. a. P(male or a college graduate) b. P(female or not a college graduate)
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Chapter 4 Probability 30. A medical research team asks 500 people whether or not they visited their physicians’ offices during the last year. The following table gives a two-way classification of the responses. Sex Male Female
Visited Physician’s Office Last Year Yes No 210 90 160 40
The team randomly selects one person from this group. Find the following probabilities. a. P(female or visited her physician’s office) b. P(male or did not visit his physician’s office) 31. The following table gives a two-way classification of 1000 couples based on whether one or both spouses work and whether or not they have children. Work Status Both Spouses Work Only One Spouse Works
Have Children Yes No 140 260 380 220
You select one couple randomly from among these 1000 couples. Find the following probabilities. a. P(both spouses work or have no children) b. P(only one spouse works or have children) 32. The probability that a lawyer is a female is .28, that a lawyer has a type A personality is .64, and that a lawyer is a female and has a type A personality is .10. Find the probability that a randomly selected lawyer is a female or has a type A personality. 33. Suppose 15% of all National Basketball Association players are over 7 feet tall, 68% weigh over 200 pounds, and 8% are over 7 feet tall and weigh over 200 pounds. If one of the NBA players is selected at random, find the probability that this player is either over 7 feet tall or weighs over 200 pounds. 34. A large company has a total of 300 female employees. Of them, 15 have a Master’s degree as the highest degree, 120 have a Bachelor’s degree, 140 have a high school diploma, and 25 have less than a high school diploma. Management selects randomly one female employee from this company. Find the probability that this employee has a Master’s degree or a high school diploma as the highest degree. 35. The probability that a person drinks at least five cups of coffee a day is .25 and the probability that a person has a high blood pressure is .10. Assuming that these two events are independent, find the probability that a person selected at random drinks less than five cups of coffee a day or has high blood pressure. 36. The probability that a family owns a house is .65 and that a family owns a camcorder is .12. Assuming that these two events are independent, find the probability that a family selected at random owns a house or does not own a camcorder. 37. A die with six sides is loaded so that the number two is twice as likely to be rolled as the numbers one, three, four, or six, and four times as likely to be rolled as the number five. Build a table that shows the probability of rolling each number, one through six. 101
38. A college student is planning to apply for a school grant. A friend of hers who works for the company that is offering the grant has told her that if she takes calculus in her freshman year and gets an A or B, she will be very strongly considered for the grant, but if she gets a C, D, or F, she will probably not be considered. If she doesn’t take calculus at all, she will remain a candidate. She estimates the following probabilities: Probability of getting the grant if she takes calculus and gets A or B: .80 Probability of getting the grant if she takes calculus and gets C, D, or F: .10 Probability of getting the grant if calculus not taken: .50 Probability of getting A or B in calculus: .60 What is the probability that she will get the grant if she takes calculus? What should she do to maximize the probability that she will get the grant? 39. Forty professors of sciences and liberal arts are asked where they would prefer to vacation if given the choice. Their grouped responses appear in the following table. Sciences Liberal Arts
Caribbean 8 10
Pacific Islands 4 5
Europe 3 5
Other 3 2
You choose one of the forty professors at random. What is the probability that the professor would choose the Caribbean? What is the probability that the professor is a sciences professor? Suppose the professor is a liberal arts professor. What is the probability that he/she would choose Europe? 40. You are watching a casino game where the player and a casino employee both roll a pair of fair dice. If the player’s roll is higher than the employee’s roll, the player wins. If the rolls are the same or the employee’s roll is higher than the player’s, the player loses. The casino employee rolls a six and a five for a total of 11. What is the probability that the player will win? 41. A pollster asks 50 residents of a certain community whether or not they favor term limits for Congressmen. Twenty of the residents are males who favor term limits, and six of the residents are females who do not favor term limits. The probability that a resident, selected at random, favors term limits is .70. How many residents are male and do not favor term limits? 42. Two types of computers (Types A and B) are used at a certain company that uses 200 computers in all. The users of each computer indicate whether they experience performance problems with their computers more than once per week. Fifty of the computers of Type A experience problems more than once per week. Seventy-five of the computers of Type B do not experience problems more than once per week. There are 120 computers of Type B at the company. Are the events ―Type A computer‖ and ―experience problems more than once per week‖ independent? Explain. 43. Can two events which are complementary be mutually exclusive? Can they be independent? Explain. 44. A statistician determines that a certain baseball player (Player A) hits a home run in 30% of the games in which he plays. The statistician also finds that if Player A hits a home run in a game, his team wins the game 80% of the time. What is the probability that the player hits a home run and his team does not win the game? 102
Chapter 4 Probability 45. Part of a life insurance underwriter’s job is to decide whether a person who applies for life insurance is an insurable risk; that is, an underwriter decides whether the person who is applying for insurance will live long enough to make the policy profitable. If the person buys the insurance and dies soon after, then the insurance company will pay out a large sum of money without having collected very much money in premiums from the person. So an underwriter is making a judgment as to whether the person is going to die soon. What concept of probability does an underwriter use? 46. You toss a coin, then a roll a die, and then toss the coin again. You are to draw a tree diagram for this experiment. Each ―leaf‖ of the tree will represent a possible outcome for the experiment. How many ―leaves‖ will the tree have?
Problem and Essay Solutions 1. Let: V = a household owns a VCR N = a household does not own a VCR a. Four possible outcomes: VV, VN, NV, NN V
S
V
VV
VN
NV
NN
PP
PB
BP
BB
N V N N b. (i) (ii) (iii) (iv)
VN, NV, VV; a compound event NN, VN, NV; a compound event VV; a simple event VN; a simple event
2. Let: P = marble selected is pink B = marble selected is black a. Four possible outcomes: PP, PB, BP, BB P
S
P B
P B B b. (i) (ii) (iii) (iv)
PB, BP, BB; a compound event PP, PB, BP; a compound event BB; a simple event BP; a simple event
3. The two properties of probability are: (i) the probability of an event lies in the range zero to 1 (ii) the sum of the probabilities of all outcomes for an experiment is 1 103
4. 124 200 = .62 5. 540 900 = .60 6. 1 5 = .20 7. 280 400 = .70 8. 660 1,200 = .55 9. a. (i) 150 400 = .375 (ii) 220 400 = .550 (iii) 60 180 = .333 (iv) 130 250 = .520 b. The events are not mutually exclusive because they can happen together. c. P(female) = .550, P(femaledoes not work) = .600. Hence, these two events are not independent. d. The complementary event is ―work‖ and its probability is .625 10. a. (i) 150 300 = .500 (ii) 30 300 = .100 (iii) 130 150 = .867 (iv) 10 30 = .333 b. Not mutually exclusive since they can happen together c. P(good) = .900, P(goodcompany A) = .933. Hence, these events are not independent. d. The complementary event is ―good‖ and its probability is .900 11. P(Female) = 75 300 = .25, P(FemaleProfessor in social sciences) = 30 90 = .333. Hence, these events are not independent. The events are not mutually exclusive because there are 30 female professors in social sciences (joint occurrence). 12. P(Male) = 18 40 = .45, P(MaleSenior) = 6 14 = .43. Therefore, these events are not independent. The events are not mutually exclusive because there are six male seniors (joint occurrence). 13. Complementary event: ―does not own a house‖ P(not A) = 1 – .68 = .32 14. Complementary event: ―not in favor of labor unions‖ P(not A) = 1 – .56 = .44 15. a. 25 150 = .167 b. 50 150 = .333 16. a. 210 500 = .420 b. 40 500 = .080 17.
a. 260 1,000 = .260 b. 280 1,000 = .380 104
Chapter 4 Probability 18. (50 250)(49 249) = .0392 19. (60 100)(59 99) = .358 20. .4 .4 = .16 21. .3 .7 = .21 22. (1 – .25) .10 = .075 23. .65 (1 – .12) = .572 24. .24 .30 = .800 25. .42 .6 = .700 26. 25 = 32 27. 10 20 6 = 1,200 28. a. (220 400) + (150 400) – (90 400) = .700 b. (250 400) + (180 400) – (120 400) = .775 29. a. (85 150) + (60 150) – (35 150) = .733 b. (65 150) + (90 150) – (40 150) = .767 30. a. (200 500) + (370 500) – (160 500) = .820 b. (300 500) + (130 500) – (90 500) = .680 31. a. (400 1,000) + (480 1,000) – (260 1,000) = .620 b. (600 1,000) + (520 1,000) – (380 1,000) = .740 32. .28 + .64 – .10 = .82 33. .15 + .68 – .08 = .75 34. (15 300) + (140 300) = .517 35. (1 – .25) + .10 – (1 – .25)(.10) = .775 36. .65 + (1 – .12) – (1 – .12)(.65) = .958 37. P(2) + P(2)/4 + 4 P(2)/2 = 1; P(2) = 4/13. Therefore, the complete table is as follows: Number Probability
1 2/13
2 4/13
3 2/13
105
4 2/13
5 1/13
6 2/13
38. If she takes calculus, her probability of getting the grant will be: P(A or B in calculus and gets grant) + P(C, D, or F in calculus and gets grant) = P(A or B)P(gets grantA or B) + P(C, D, or F)P(gets grantC, D, or F) = (.60)(.80) + (.40)(.10) = .48 + .04 = .52 > .50 (probability of getting grant if calculus not taken) So if she takes calculus, her probability of getting the grant will be higher than if she doesn’t take it. Therefore, she should take calculus. 39. 18 professors chose the Caribbean, so the first (marginal) probability is 18 40 = .45 18 of the professors are science professors, so the second (marginal) probability is also .45 22 of the professors are in liberal arts, and five of them chose Europe, so the third (conditional) probability is 5 22 = .227 40. The player will win only if he/she rolls a twelve, for a probability of 1/36. 41. The number of residents that favor term limits is .70 50 = 35. Twenty of these are males, so 15 are females. Six residents are female and do not favor term limits, so 50 – 20 – 15 – 6 = 9 residents are males who do not favor term limits. 42. P(Type B computer) = 120 200 = .60, so P(Type A computer) = .40 There are 120 Type B computers, and 75 of them do not experience any problems, so 45 of them do. That makes 95 computers that experience problems. Therefore, P(Type A computerexperience problems) = 50 95 = .526 .40. So the two events are not independent. 43. Complementary events are, by definition, mutually exclusive. For events A and B to be independent, we must have P(A) = P(AB) = P(A and B) P(B). But if A and B are complementary, then P(A and B) = 0, so we cannot have P(A) = P(AB). Therefore, complementary events are never independent. 44. It is given that P(home run) = .30 and P(win gamehome run) = .80. Therefore, P(lose gamehome run) = .20, and P(home run and lose game) = P(lose gamehome run)P(home run) = .06. 45. This is a subjective probability because the probability of death is a judgment call. 46.
Since each ―leaf‖ represents an outcome, there will be 2 6 2 = 24 ―leaves‖ depicted on the tree diagram.
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